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Constructions of k-regular maps using finite local schemes

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2019

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Buczyński, Jarosław
Januszkiewicz, Tadeusz
Jelisiejew, Joachim

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https://doi.org/10.4171/JEMS/873
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Journal of the European Mathematical Society. European Mathematical Society (EMS). 2019, 21(6), pp. 1775-1808. ISSN 1435-9855. eISSN 1435-9863. Available under: doi: 10.4171/JEMS/873

Zusammenfassung

A continuous map Rm→RN or Cm→CN is called k-regular if the images of any k points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N, but there are very few results on the existence of such maps for particular values m and k. Using methods of algebraic geometry we construct k-regular maps. We relate the upper bounds on N with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k≤9, and we provide explicit examples for k≤5. We also provide upper bounds for arbitrary m and k.

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510 Mathematik

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k-regular embeddings, secants, punctual Hilbert scheme, finite Gorenstein schemes

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ISO 690BUCZYŃSKI, Jarosław, Tadeusz JANUSZKIEWICZ, Joachim JELISIEJEW, Mateusz MICHALEK, 2019. Constructions of k-regular maps using finite local schemes. In: Journal of the European Mathematical Society. European Mathematical Society (EMS). 2019, 21(6), pp. 1775-1808. ISSN 1435-9855. eISSN 1435-9863. Available under: doi: 10.4171/JEMS/873
BibTex
@article{Buczynski2019Const-52566,
  year={2019},
  doi={10.4171/JEMS/873},
  title={Constructions of k-regular maps using finite local schemes},
  number={6},
  volume={21},
  issn={1435-9855},
  journal={Journal of the European Mathematical Society},
  pages={1775--1808},
  author={Buczyński, Jarosław and Januszkiewicz, Tadeusz and Jelisiejew, Joachim and Michalek, Mateusz}
}
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    <dcterms:abstract xml:lang="eng">A continuous map R&lt;sup&gt;m&lt;/sup&gt;→R&lt;sup&gt;N&lt;/sup&gt; or C&lt;sup&gt;m&lt;/sup&gt;→C&lt;sup&gt;N&lt;/sup&gt; is called k-regular if the images of any k points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N, but there are very few results on the existence of such maps for particular values m and k. Using methods of algebraic geometry we construct k-regular maps. We relate the upper bounds on N with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k≤9, and we provide explicit examples for k≤5. We also provide upper bounds for arbitrary m and k.</dcterms:abstract>
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